BASIC ALGEBRAIC OPERATIONS Unit 13 BASIC ALGEBRAIC OPERATIONS

ADDITION Only like terms can be added. The addition of unlike terms can only be indicated Procedure for adding like terms: Add the numerical coefficients, applying the procedure for addition of signed numbers Leave the variables unchanged 6y + (–5y) = 1y = y Ans –13ab + (–11ab) = –24ab Ans

ADDITION Procedure for adding expressions that consist of two or more terms: Group like terms in the same column Add like terms and indicate the addition of the unlike terms Add: 5y + (–3x) + 6x2y and (–4x) + (–2y) + (–2x2y) Add the like terms and indicate the addition of the unlike terms

SUBTRACTION Just as in addition, only like terms can be subtracted Each term of the subtrahend is subtracted following the procedure for subtraction of signed numbers Subtract: (7x2 + 7xy – 15y2) – (–8x2 + 5xy – 10y2) Change the sign of each term in the subtrahend and follow the procedure for addition of signed numbers

MULTIPLICATION In multiplication, the exponents of the literal factors do not have to be the same to multiply the values Procedure for multiplying two or more terms: Multiply the numerical coefficients, following the procedure for multiplication of signed numbers Add the exponents of the same literal factors Show the product as a combination of all numerical and literal factors Multiply: (–4)(5x)(–6x2y)(7xy)(–2y3) Multiply all coefficients and add exponents of the same literal factors = (–4)(5)(–6)(7)(–2)(x1 + 2 + 1)(y1 + 1 + 3) = –1680x4y5 Ans

MULTIPLICATION Procedure for multiplying expressions that consist of more than one term within an expression: Multiply each term of one expression by each term of the other expression Combine like terms b. (2a – 3b)(5a + 2b) = (2a)(5a) + (2a)(2b) + (–3b)(5a) + (–3b)(2b) a. 2x(3x2 + 2x – 5) = 2x(3x2) + 2x(2x) + (2x)(–5) = 10a2 + 4ab – 15ab – 6b2 = 6x3 + 4x2 – 10x Ans = 10a2 – 11ab – 6b2 Ans

DIVISION Procedure for dividing two terms: Divide the numerical coefficients following the procedure for division of signed numbers Subtract the exponents of the literal factors of the divisor from the exponents of the same letter factors of the dividend Combine numerical and literal factors

DIVISION Divide: (40a3b4c5)  (–4ab2c3) = –10a2b2c2 Ans

DIVISION Procedure for dividing when the dividend consists of more than one term: Divide each term of the dividend by the divisor, following the procedure for division of signed numbers Combine terms Divide:

(x + y)2 = (x + y)(x + y) = x2 + xy + xy + y2 POWERS Procedure for raising a single term to a power: Raise the numerical coefficients to the indicated power following the procedure for powers of signed numbers Multiply each of the literal factor exponents by the exponent of the power to which it is raised Combine numerical and literal factors Procedure for raising two or more terms to a power: Apply the procedure for multiplying expressions that consist of more than one term (FOIL) (2x2y3)2 = 22(x2)2(y3)2 = 4x4y6 Ans (x + y)2 = (x + y)(x + y) = x2 + xy + xy + y2 = x2 + 2xy + y2 Ans

ROOTS Procedures for extracting the root of a term: Solve: Determine the root of the numerical coefficient following the procedure for roots of signed numbers The roots of the literal factors are determined by dividing the exponent of each literal factor by the index of the root Combine the numerical and literal factors Solve: = –2a2b3c Ans

REMOVAL OF PARENTHESES Procedure for removal of parentheses preceded by a plus sign: Remove the parentheses without changing the signs of any terms within the parentheses Combine like terms 7x + (–4x + 3y – 2) = –7x – 4x + 3y – 2 = –11x + 3y – 2 Ans Procedure for removal of parentheses preceded by a minus sign: Remove the parentheses and change the sign of each term within the parentheses 9a – (–4a + 2b – 6) = 9a + 4a – 2b + 6 = 13a – 2b + 6 Ans

COMBINED OPERATIONS Simplify: [3x – x + (x2y3)2]2 Expressions that consist of two or more different operations are solved by applying the proper order of operations Simplify: 15x – 4(–2x) + x = 15x + 8x + x = 23x + x = 24x Ans Simplify: [3x – x + (x2y3)2]2 = [3x – x + x4y6]2 = [2x + x4y6]2 = (2x + x4y6)(2x + x4y6) = 4x2 + 2x5y6 + 2x5y6 + x8y12 = 4x2 + 4x5y6 + x8y12 Ans

BINARY NUMERATION SYSTEM The binary number system uses only the two digits 0 and 1. These two digits are the building blocks for the binary code that is used to represent data and program instructions for computers Place values for binary numbers are shown below: Remember this can continue as far as needed in either direction

EXPRESSING BINARY NUMBERS AS DECIMAL NUMBERS We use a subscript 2 to show that a number is binary Express the binary number 1012 as a base 10 decimal number: 1012 = 1(22) + 0(21) + 1(20) = 4 + 0 + 1 = 510 Ans Express the decimal number 1910 as a binary number: The largest power of two that will divide into 19 is 24 or 16 19 = 1(24) + 0(23) + 0(22) + 1(21) + 1(20) = 100112 Ans

PRACTICE PROBLEMS Perform the indicated operations and simplify: 6a + 7a + 9b (–3xy) + 4x + (–5xy2) + 5xy + (–7x) –3.07ab + 7.69c + (–5.76ab) + 9d + (–11.2c) 1/2x + (–2/3y) + 1/4z + (–1/3z) + 2/3x 4x2y + (–5xy2) + 7xy2 + (–2x2y) 7a – 3a –10x – (–20x) (3y2 – 4z) – (–2y2 + 4z) –1 1/2ab – 1 2/3ab (–2.04t2 + 7.6t – 7) – (3t2 – 6.7t – 4)

PRACTICE PROBLEMS (Cont) (3ab)(–4a2b2) (–1/2x)(–1/3y2)(1/4x3) (a – b)(a – b) (2x2 – 3y)(–3x2 + 2y) 16y2  4y 1 1/3 a2b3  2/3ab (2.4x3y3 + 4.8x2y2 – 24x)  1.2x (x2y)3 (–2.1a2b3)2 (2/3 x3y3z2)3 (–2m2n + p3)2

PRACTICE PROBLEMS (Cont) 22. 23. 24. 25. 2x – (x – 2y) 26. –(x + y – z) – (x2 – y2 + z) 27. 15 – (ab – a2b – b) – 4 + (ab – b) 28. (18a4b2)  (3a2b) – b3(b2) 29. 5(2x – y)2 – (x2 – y2)

PROBLEM ANSWER KEY 13a + 9b 2xy + (–3x) + (–5xy2) –8.83ab + (–3.51c) + 9d 7/6x + (–2/3y) + (–1/12z) 2x2y + 2xy2 4a 10x 5y2 – 8z –3 1/6ab –5.04t2 + 14.3t – 3

PROBLEM ANSWER KEY (Cont) –12a3b3 1/24x4y2 a2 – 2ab + b2 –6x4 + 13x2y – 6y2 4y 2ab2 2x2y3 + 4xy2 – 20 x6y3 4.41a4b6 8/27x9y9z6 4m4n2 – 4m2np3 + p6

PROBLEM ANSWER KEY (Cont) 11a3b2c – 4x4y3 x + 2y –x2 + y2 – x – y 11 + a2b 6a2b – b5 19x2 – 20xy + 6y2